Let X be a smooth algebraic surface. A foliation F on X is, roughly
speaking, a subline bundle T_F of the tangent bundle of X. The dual of
T_F is called the canonical bundle of the foliation K_F.
In the last few years birational methods have been successfully used in order to study foliations.
More precisely, geometric properties of the foliation are translated into properties of the canonical bundle of the foliation.
One of the most important invariants describing the properties of a
line bundle L is its Kodaira dimension kod(L), which measures the growth
of the global sections of L and its tensor powers.
The Kodaira dimension of a foliation F is defined as the Kodaira dimension of its canonical bundle kod(K_F).
In their fundamental WORKS, Brunella and McQuillan give a
classfication of foliations on surfaces on the model of Enriques-Kodaira
classification of surfaces.

The next step is the study of the behaviour of families of foliations.
Brunella proves that, for a family of foliations (X_t, F_t) of
dimension one on surfaces, satisfying certain hypotheses of regularity,
the Kodaira dimension of the foliation does not depend on t.
By analogy with Siu's Invariance of Plurigenera, it is natural to
ask whether for a family of foliations (X_t, F_t) the dimensions of
global sections of the canonical bundle and its powers depend on t.
In this talk we will discuss to which extent an Invariance of
Plurigenera for foliations is true and under which hypotheses on the
family of foliations it holds.